In order to find the place for the support of the table, we need to find the balancing point. In a triangle, the balancing point is a of the triangle. Thus, we need to locate the centroid of the triangle. First, using the given coordinates, we will draw the on a coordinate plane.
Let's recall that a centroid of a triangle is point of intersection of the triangle . Thus, to locate the centroid, we can draw the medians of our triangle and find their point of concurrency. The other way to find the centroid is to use the . Let's review what it states.
The medians of a triangle inersectat a point called the centroid that istwo thirds of the distance from each vertexto the midpoint of the opposite side.
First, we can draw a median from the vertex with the coordinates
(3,6). To do this, we need to know the coordinates of the of the opposite side. Let's find them substituting
(7,10) and
(5,2) into the .
M(2x1+x2,2y1+y2)
M(27+5,210+2)
M(212,212)
M(6,6)
Now, we can plot
M(6,6) on the coordinate plane and draw the median. Let's also name the vertices of the triangle for the following explanation to be easier.
According to the above theorem, the centroid of
△ABC is two thirds of the distance from
A to
M. If we call the centroid
D, the following equality is true.
AD=32AM
To find
AM, let's analyze the diagram. We can see that
A(3,6) and
M(6,6) have the same
y-coordinate,
6. Hence, to find the measure of
AM, we can calculate the difference between their
x-coordinates.
AM=6−3=3
Now, let's substitute
3 for
AM into our equation and solve it for
AD.
AD=32AM
AD=32(3)
AD=32⋅3
AD=2
Since the measure of
AD is
2, by adding
2 to the
x-coordinate of
A(3,6) we can find the coordinates of the centroid
D.
D(3+2,6) ⇒ D(5,6)
Therefore, the coordinates of the centroid of the triangle are
(5,6).